In the Black Scholes setting, here is how my understanding is of how we derive the PDE for the value of an option.
- We assume that the price of the option is Markovian in our state variable $S_t$. Then we can use Ito's formula to get the SDE of our option process.
We form a portfolio of the underlying and the option. Using Ito, we can know get a SDE for our portfolio process.
Using the SDE of the portfolio we make it independent of $dS_t$ (and thus also $dW_t$, i.e. it is locally riskless) by choosing our portfolio weights appropiately.
Using arbitrage arguments, we know then that the remaining $dt$-term in the portfolio process must equal $r(t)$.
- This condition is the Black Scholes PDE.
- Using a Feynman-Kac representation, we can convert this PDE representation to a discounted expectation but under a "different" (risk-neutral) measure.
Now, for a non-Black Scholes setting (e.g. stochastic volatility), how must this procedure be modified to produce the PDE for the option AND the risk-neutral expectation? In particular, I have 2 questions:
- Do we still assume that the price of the option is Markovian in $S_t$ but now also in $\sigma_t$, the volatility?
- When forming the portfolio, I need to ensure that the $dS_t$ and the $d\sigma_t$ terms in the SDE of the portfolio process disappear, so that we have a locally riskless portfolio. But how can I do this? If I am investing in the option + the underlying, I have 2 variables but 3 equations.